We present an analysis of the AGN broad-line regions of 6 powerful radio galaxies at z>~2 (HzRGs) with rest-frame optical imaging spectroscopy obtained at the VLT. All galaxies have luminous (L(H-alpha)=few x 10^44 erg s^-1), spatially unresolved H-alpha line emission with FWHM>= 10,000 km s^-1 at the position of the nucleus, suggesting their AGN are powered by supermassive black holes with masses of few x 10^9 M_sun and accretion luminosities of a few percent of the Eddington luminosity. In two galaxies we also detect the BLRs in H-beta, suggesting relatively low extinction of A_V~1 mag, which agrees with constraints from X-ray observations. By relating black hole and bulge mass, we find a possible offset towards higher black-hole masses of at most ~0.6 dex relative to nearby galaxies at a given host mass, although each individual galaxy is within the scatter of the local relationship. If not entirely from systematic effects, this would then suggest that the masses of the host galaxies have increased by at most a factor ~4 since z~2 relative to the black-hole masses, perhaps through accretion of satellite galaxies or because of a time lag between star formation in the host galaxy and AGN fueling. We also compare the radiative and mechanical energy output (from jets) of our targets with predictions of recent models of "synthesis" or "grand unified" AGN feedback, which postulate that AGN with similar radiative and mechanical energy output rates to those found in our HzRGs may be nearing the end of their period of active growth. We discuss evidence that they may reach this stage at the same time as their host galaxies.